Some plays, like the Tempest, were only printed in the First Folio of 1623. Other plays, like Romeo and Juliet were printed in both quarto and the First Folio. Pericles saw five quarto printings (the most of any Shakespeare play), and then wasn't included in the First Folio at all!
Hamlet is a unique problem in that it has three primary source texts, and the three source texts are profoundly different. The first quarto printed in 1603 (Q1 or "the bad quarto") is half as long as the other two texts. Many of the characters have different names. The second quarto of 1604 (Q2) was almost twice as long as Q1 and has a title page claiming it had been "newly imprinted and enlarged to almost as much again as it was, according to the true and perfect Coppie".
The First Folio of 1623 (F1) is about the same length as Q2, but again it has significant differences. For example Hamlet's famous comment to Horatio reads in F1 "There are more things in Heauen and Earth, Horatio, / Then are dream't of in our Philosophy." Q1 and Q2 both say "in your philosophy". The difference of a single letter can totally change the gist of a line. F1 contains some 70 lines not present in Q2 while omitting about 230 lines that appear in Q2. There are also extensive differences in punctuation between the texts.
Editors of Shakespeare's plays for centuries would update, tweak, or even blatantly alter Shakespeare's work. For example a man named Nahum Tate reworked King Lear and gave it a happy ending. This version was acted almost exclusively for a couple centuries. Most editorial changes were less dramatic, consisting of picking between alternative spellings, word choices, or variant punctuations.
Until the mid 20th century, editors continued freely used the various source texts for Hamlet, mixing and matching between them. This led to what are now called "conflated texts". A conflated text indiscriminately picks and chooses between the three different sources, hoping to re-create the idealized "true" Hamlet that Shakespeare intended. In the 1970s a group of Shakespeare scholars including Stanley Wells, Gary Taylor, and Michael Warren argued that this conflation was illegitimate. The different source texts aren't necessarily variations on a single play, but they can actually represent distinctly different plays! Their prime examples were Hamlet and King Lear. King Lear is particularly problematic because, they argued, in one source text the play essentially has a happier ending where Lear thinks Cordelia lives, while the other version Lear dies with no hope and the ending is more nihilistic. This isn't just a tweak but an different interpretation.
Today, both conflated and deconflated texts of Shakespeare's work are available. The most recent Arden Shakespeare edition of Hamlet from 2006 prints all three source texts in full and allows the reader to pick between them. King Lear has also been printed in its separate original source texts.
No text of Hamlet is inherently better than any other, and there are merits for each source text and for the endless conflated texts editors have published over the centuries. The text we are using for the Truepenny Players production derives from OpenSource Shakespeare, a conflated text.
A basic knowledge of the different source texts allows you to investigate problematic or interesting words, lines, or passages. Sometimes an editor has actually made a significant choice for you without your knowledge! Like with the "your" and "our" variation mentioned above, simple changes can be massively important. With the internet today, actors and directors are fully capable of making these decisions for themselves.
Quarto 1 [the 'Bad Quarto'] facsimile (1603)
Quarto 2 facsimile (1604)
First Folio (1623) text
First Folio (1623) facsimile
Enfolded Hamlet (compare Q2 and F1)